There are probably as many variant combinations of motivations as there are people buying vintage gear, but among the main motivators are probably:
(1) SONIC VALUE. Better sound for less money. If you are buying yesteryear's high-end gear, you're getting great sound, but typically you're paying a lot less than you'd have to, to get anything close to it in new gear. Sure, you can overpay for a lemon that will need so much work done that it won't be a bargain, or you might buy something that won't sound all that good no matter what you do to it... but assuming you have some idea what you are doing (or at least check on AK first!

), you CAN get great sonic bang-for-your-buck buying vintage.
(2) BUILD QUALITY/DURABILITY: Yes, there is new gear that is made with high-end components, thick slabs of aluminum for faceplates, etc... but that stuff costs a lot and is only found in specialty stores. Most of what's out there in the mass market today is plastic, made with surface-mounted components and generally not made to last for too long. Good vintage gear, with a minor overhaul every decade or two, will last for decades. Modern stuff usually won't last for even one decade. Except for the pricey boutique stuff, they don't make 'em like they used to. Even in the few areas where the sound quality has improved, like headphones, the quality of the frames, cups, etc... has cheapened and become less durable.
(3) MORE SCOPE TO "PLAY": This is a natural outgrowth of #1 above. Since vintage gear can sometimes be bought so cheaply at garage sales, thrift shops, etc..., many hobbyists can afford to try this, try that, and try the other thing, without risking too much financially. They can set up multiple systems, one for each room of the house if they like. Try "churning through" as much new gear at full retail, and you'd better be rich, or else you'll soon be bankrupt.
(4) FORMATS AND TITLES: Okay, it is possible to buy a new turntable and new records today... but it was difficult for a while, and many titles originally released on vinyl are still only available in old, original pressings. And if you are fond of magnetic tape in reel or cassette format, or even more obscure formats like DAT, DCC, wire, laserdisc, elcassette, etc..., then you have no choice but to go vintage.
(5) NOSTALGIA: Many of the people active on AK are from the "baby boomer" generation, old enough to remember when the sought-after vintage gear was not vintage, but brand new. A lot of us pored over magazine or catalog ads, visited showrooms, and dreamed of owning this or THAT piece of gear... but those were only dreams, way beyond the dreams of average youth. Today, that exact same gear is affordable; our incomes have increased, and the prices of those pieces have dropped. Collecting vintage gear today can be a means of making yesterday's dreams really come true, at last.
(6) VISUAL APPEAL: A lot of us prefer the looks of, say, an SX-1980 with it's silver face, wood slab sides and cool black heatsinks to a modern plastic-cased radio. We prefer a turntable with a wooden plinth and silver or black metal parts, to one with a spray-painted plinth, clear plastic platter and black plastic tonearm. Vintage gear fits better with vintage interiors, and has more appeal for those who decorate with antique furniture.
(7) THE THRILL OF THE HUNT: For some, the act of just going out and seeking gear, never knowing what will show up THIS time at the thrift (garage sale, wherever), is fun, like a treasure hunt. The anticipation of "what if THIS is the time when that original Marantz tube unit or Tannoy Westminsters are sitting there..." creates excitement, and the occasional "scroe" brings a thrill that is probably akin to a gambler's score... but no money has to be risked, to play this game! The roots of this motivation probably like in our genetics, the result of our hunter-gatherer ancestor's need to "score" food on a regular basis. This is hunting, without having to kill anything (except time

).
There are other reasons why people might collect vintage gear, but I think these are among the most common.